DOI:https://doi.org/10.15407/kvt189.03.029
Kibern. vyčisl. teh., 2017, Issue 3 (189), pp.
Zhiteckii L.S., PhD (Engineering), Acting Head of the Department of Intelligent Automatic Systems
e-mail: leonid_zhiteckii@i.ua
Solovchuk K.Yu., Postgraduate Student
e-mail: solovchuk_ok@mail.ru
International Research and Training Center for Information Technologies and Systems of the NAS of Ukraine and Ministry of Education and Science of Ukraine,
Acad. Glushkova av., 40, Kiev, 03680, Ukraine
DISCRETE-TIME STEADY-STATE CONTROL OF INTERCONNECTED SYSTEMS BASED ON PSEUDOINVERSION CONCEPT
Introduction. The problem of controlling interconnected systems subjected to arbitrary unmeasurable disturbances remains actual up to now. It is important problem from both theoretical and practical points of view. During the last decades, the internal model control principle becomes popular among other methods dealing with an improvement of the control system. A perspective modification of the internal model control principle is the so-called model inverse approach. Unfortunately, the inverse model approach is quite unacceptable if the systems to be controlled are square but singular or if they are nonsquare. It turned out that the so-called pseudoinverse (generalized inverse) model approach can be exploited to cope with the noninevitability of singular square and also nonsquare system.
The purpose of the paper is to generalize the results obtained by the authors in their last works which are related to the asymptotic properties of the pseudoinverse model-based method for designing an efficient steady-state control of interconnected systems with uncertainties and arbitrary bounded disturbances and also to present some new results.
Results. In this paper, the main effort is focused on analyzing the asymptotic properties of the closed-loop systems containing the pseudoinverse model-based controllers. In the framework of the pseudoinversion concept, new theoretical results related to the asymptotic behavior of these systems are obtained. Namely, in the case of nonsingular gain matrices with known elements, the upper bounds on the ultimate norms of output and control input vectors are found. Next, in the case of nonsquare gain matrices whose elements are also known, the asymptotic behavior of the feedback control systems designed on the basis of pseudoinverse approach are studied. Further, the sufficient conditions guaranteeing the boundedness of the output and control input signals for the linear and certain class of nonlinear interconnected systems in the presence of uncertainties are derived.
Conclusion. It has been established that the pseudoinverse model-based concept can be used as a unified concept to deal with the steady-state regulation of the linear interconnected discrete-time systems and of some classes of nonlinear interconnected systems with possible uncertainties in the presence of arbitrary unmeasured but bounded disturbances.
Keywords: discrete time, feedback, pseudoinversion, interconnected systems, optimality, stability, uncertainty.
REFERENCES
1 Davison E. The output control of linear time-invariant multivariable systems with unmeasurable arbitrary disturbances. IEEE Trans. Autom. Contr., 1972, vol. AC-17, no. 5, pp. 621–631.
https://doi.org/10.1109/TAC.1972.1100084
2 Liu C., Peng H. Inverse-dynamics based state and disturbance observers for linear time-invariant systems. ASME J. Dyn Syst., Meas. and Control, 2002, vol. 124, no. 5, pp. 376–381.
3 Lyubchyk L. M. Disturbance rejection in linear discrete Multivariable systems: inverse model approach. Prep. 18th IFAC World Congress, Milano, Italy, 2011, pp. 7921–7926.
https://doi.org/10.3182/20110828-6-IT-1002.02121
4 Skogestad S., Postlethwaite I. Multivariable Feedback Control. UK, Chichester: Wiley, 1996.
5 Freudenberg J. and Middleton R. Properties of single input, two output feedback systems. Int. J. Control, 1999, vol. 72, no. 16, pp. 1446–1465.
https://doi.org/10.1080/002071799220100
6 Francis B., Wonham W. The internal model principle of control theory. Automatica, 1976, vol. 12, no. 5, pp. 457–465.
https://doi.org/10.1016/0005-1098(76)90006-6
7 Brockett R. W. The invertibility of dynamic systems with application to control. Ph. D. Dissertation, Case Inst. of Technology, Cleveland, Ohio, 1963.
8 Sain M. K., Massey J. L. Invertibility of linear time-invariant dynamical systems. IEEE Trans. Autom. Contr.,1969, vol. AC-14, no. 2, pp. 141–149, Apr. 1969.
9 Silverman L. M. Inversion of multivariable linear systems. IEEE Trans. Autom. Contr., 1969, vol. AC-14, no. 3, pp. 270–276.
https://doi.org/10.1109/TAC.1969.1099169
10 Lovass-Nagy V., Miller J. R., Powers L. D. On the application of matrix generalized inversion to the construction of inverse systems. Int. J. Control, 1976, vol. 24, no. 5, pp. 733–739.
https://doi.org/10.1080/00207177608932859
11 Seraji H. Minimal inversion, command tracking and disturbance decoupling in multivariable systems. Int. J. Control, 1089, vol. 49, no. 6, pp. 2093–2191.
12 Marro G., Prattichizzo D., Zattoni E. Convolution profiles for right-inversion of multivariable non-minimum phase discrete-time systems. Automatica, 2002, vol. 38, no. 10, pp. 1695–1703.
https://doi.org/10.1016/S0005-1098(02)00088-2
13 Pukhov G. E., Zhuk K. D. Synthesis of Interconnected Control Systems via Inverse Operator Method. Kiev: Nauk. dumka, 1966 (in Russian).
14 Lee T., Adams G., Gaines W. Computer Process Control: Modeling and Optimization. New York: Wiley, 1968.
15 Skurikhin V. I., Procenko N. M., Zhiteckii L. S. Multiple-connected systems of technological processes control with table of objects. Proc. IFAC Third Multivariable Tech. Systems Symp., Manchester, U.K., 1974, pp. S 35-1 – S 35-4.
https://doi.org/10.1016/S1474-6670(17)69194-8
16 Katkovnik V. Ya., Pervozvansky A. A. Methods for the search of extremum and the synthesis problems of multivariable control systems. Adaptivnye Avtomaticheskie Sistemy, Moscow: Sov. Radio, pp. 17–42, 1973 (in Russian).
17 Skogestad S., Morari M., Doyle J. Robust control of ill-conditioned plants: high purity distillation. IEEE Trans. Autom. Contr., 1988, vol. 33, no. 12, pp. 1092–1105.
https://doi.org/10.1109/9.14431
18 Skurikhin V. I., Zhiteckii L. S., Solovchuk K. Yu. Control of interconnected plants with singular and ill-conditioned transfer matrices based on pseudo-inverse operator method. Upravlyayushchye sistemy i mashiny, 2013, no. 3, pp. 14-20, 29 (in Russian).
19 Zhiteckii L. S., Azarskov V. N., Solovchuk K. Yu., Sushchenko O. A. Discrete-time robust steady-state control of nonlinear multivariable systems: a unified approach. Proc. 19th IFAC World Congress, Cape Town, South Africa, 2014, pp. 8140–8145.
https://doi.org/10.3182/20140824-6-ZA-1003.01985
20 Skurikhin V. I., Gritsenko V. I., Zhiteckii L. S., Solovchuk K. Yu. Generalized inverse operator method in the problem of optimal controlling linear interconnected static plants. Dopovidi NAN Ukrainy, no. 8, pp. 57–66, 2014 (in Russian).
21 Albert A. Regression and the Moore-Penrose Pseudoinverse. New York: Academic Press, 1972.
22 Zhiteckii L. S., Skurikhin V. I. Adaptive Control Systems with Parametric and Nonparametric Uncertainties. Kiev: Nauk. dumka, 2010 (in Russian).
23 Lancaster P., Tismenetsky M. The Theory of Matrices: 2nd ed. With Applications. N.Y.: Academic Press, 1985.
Received 17.02.2017